A120019 Square table, read by antidiagonals, of self-compositions of A120010.
1, 1, 1, 1, 2, 1, 1, 3, 4, 2, 1, 4, 9, 10, 6, 1, 5, 16, 30, 32, 18, 1, 6, 25, 68, 114, 116, 53, 1, 7, 36, 130, 312, 480, 440, 158, 1, 8, 49, 222, 710, 1536, 2157, 1708, 481, 1, 9, 64, 350, 1416, 4070, 8000, 10092, 6760, 1491, 1, 10, 81, 520, 2562, 9348, 24365, 43472, 48525
Offset: 1
Examples
Square table begins: 1, 1, 1, 2, 6, 18, 53, 158, 481, 1491, ... 1, 2, 4, 10, 32, 116, 440, 1708, 6760, 27232, ... 1, 3, 9, 30, 114, 480, 2157, 10092, 48525, 238143, ... 1, 4, 16, 68, 312, 1536, 8000, 43472, 243808, 1400448, ... 1, 5, 25, 130, 710, 4070, 24365, 151330, 968785, 6355795, ... 1, 6, 36, 222, 1416, 9348, 63768, 448188, 3234216, 23875296, ... 1, 7, 49, 350, 2562, 19236, 148085, 1167488, 9409645, 77367087, ... 1, 8, 64, 520, 4304, 36320, 312512, 2740672, 24476800, 222358528, ... 1, 9, 81, 738, 6822, 64026, 610245, 5906502, 58033953, 578488563, ... 1, 10, 100, 1010, 10320, 106740, 1117880, 11855660, 127313320, ... Successive self-compositions of F(x), the g.f. of A120010, begin: F(x) = x + x^2 + x^3 + 2x^4 + 6x^5 + 18x^6 + 53x^7 + 158x^8 +... F(F(x)) = x + 2x^2 + 4x^3 + 10x^4 + 32x^5 + 116x^6 + 440x^7 +... F(F(F(x))) = x + 3x^2 + 9x^3 + 30x^4 + 114x^5 + 480x^6 + 2157x^7 +... F(F(F(F(x)))) = x + 4x^2 + 16x^3 + 68x^4 + 312x^5 + 1536x^6 +... F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + 130x^4 + 710x^5 + 4070x^6 +... F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 + 222x^4 + 1416x^5 + 9348x^6 +...
Programs
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PARI
{T(n,k)=sum(j=1, k, binomial(2*k-2*j, k-j)/(k-j+1)* sum(i=1, j,(-1)^(j-i)*binomial(k-j+i, j-i)*binomial(k-j+i-1, i-1)*n^(i-1)))}
Formula
T(n, k) = Sum_{j=1..k}Catalan(k-j)*[Sum_{i=1..j}(-1)^(j-i)*n^(i-1)*C(k-j+i, j-i)*C(k-j+i-1, i-1)]; Also, T(n, k) = Sum_{j=0..k-1}n^j*[Sum_{i=j..k-1}(-1)^(i-j)*Catalan(k-i-1)*C(k-i+j, i-j)*C(k-i+j-1, j)]; where Catalan(n) = A000108(n) = C(2n, n)/(n+1).
Comments